Dimensionality problem

Pratt L R 1986 A statistical method for identifying transition states in high dimensional problems J. Chem. Phys. 85 5045-8... 

The result of this approximation is that each mode is subject to an effective average potential created by all the expectation values of the other modes. Usually the modes are propagated self-consistently. The effective potentials governing die evolution of the mean-field modes will change in time as the system evolves. The advantage of this method is that a multi-dimensional problem is reduced to several one-dimensional problems. 

Components of the governing equations of the process can be decoupled to develop a solution scheme for a three-dimensional problem by combining one- and two-dimensional analyses. 

Isoparametric mapping described in Section 1.7 for generating curved and distorted elements is not, in general, relevant to one-dimensional problems. However, the problem solved in this section provides a simple example for the illustration of important aspects of this procedure. Consider a master element as is shown in Figure 2.23. The shape functions associated with this element are... 

Note that in the one-dimensional problem illustrated here the Jacobian of coordinate transformation is simply expressed as dx7d and therefore... 

As the number of elements in the mesh increases the sparse banded nature of the global set of equations becomes increasingly more apparent. However, as Equation (6,4) shows, unlike the one-dimensional examples given in Chapter 2, the bandwidth in the coefficient matrix in multi-dimensional problems is not constant and the main band may include zeros in its interior terms. It is of course desirable to minimize the bandwidth and, as far as possible, prevent the appearance of zeros inside the band. The order of node numbering during... 

The hydrogen atom is a three-dimensional problem in which the attractive force of the nucleus has spherical symmetr7. Therefore, it is advantageous to set up and solve the problem in spherical polar coordinates r, 0, and three parts, one a function of r only, one a function of 0 only, and one a function of [Pg.171]

B. Expresses the eigenvalue of the original equation as a sum of eigenvlaues (whose values are determined via boundary conditions as usual) of the lower-dimensional problems. 

We begin the mathematical analysis of the model, by considering the forces acting on one of the beads. If the sample is subject to stress in only one direction, it is sufficient to set up a one-dimensional problem and examine the components of force, velocity, and displacement in the direction of the stress. We assume this to be the z direction. The subchains and their associated beads and springs are indexed from 1 to N we focus attention on the ith. The absolute coordinates of the beads do not concern us, only their displacements. 

The model describing interaction between two bodies, one of which is a deformed solid and the other is a rigid one, we call a contact problem. After the deformation, the rigid body (called also punch or obstacle) remains invariable, and the solid must not penetrate into the punch. Meanwhile, it is assumed that the contact area (i.e. the set where the boundary of the deformed solid coincides with the obstacle surface) is unknown a priori. This condition is physically acceptable and is called a nonpenetration condition. We intend to give a mathematical description of nonpenetration conditions to diversified models of solids for contact and crack problems. Indeed, as one will see, the nonpenetration of crack surfaces is similar to contact problems. In this subsection, the contact problems for two-dimensional problems characterizing constraints imposed inside a domain are considered. 

These time steps are smaller than for one-dimensional problems. For three dimensions, the limit is... 

To avoid such small time steps, which become smaller as Ax decreases, an implicit method could be used. This leads to large, sparse matrices rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful (Ref. 221). This reduces a problem on an /i X n grid to a series of 2n one-dimensional problems on an n grid. 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors... 

We have been discussing a class of penetration problems that are accurately modeled by two-dimensional calculations. There are many three-dimensional problems, however, that can be well approximated by two-dimensional analyses, and the greatly reduced computer memory and time requirements for such calculations make them attractive alternatives for scoping studies, or for parameter sensitivity studies. Although good quantitative predictions may not be obtained with such approximations, the calculations can be expected to reveal trends and qualitative results that will carry over to the full three-dimensional problem. 

Computational methods have played an exceedingly important role in understanding the fundamental aspects of shock compression and in solving complex shock-wave problems. Major advances in the numerical algorithms used for solving dynamic problems, coupled with the tremendous increase in computational capabilities, have made many problems tractable that only a few years ago could not have been solved. It is now possible to perform two-dimensional molecular dynamics simulations with a high degree of accuracy, and three-dimensional problems can also be solved with moderate accuracy. 

Similar treatments can be used for all sorts of two-dimensional problems for calculating the plastic collapse load of structures of complex shape, and for analysing metal-working processes like forging, rolling and sheet drawing. 

A number of empirical tunneling paths have been proposed in order to simplify the two-dimensional problem. Among those are MEP [Kato et al. 1977], sudden straight line [Makri and Miller 1989], and the so-called expectation-value path [Shida et al. 1989]. The results of these papers are hard to compare because slightly different PES were used. As to the expectation-value path, it was constructed as a parametric line q(Q) on which the vibration coordinate q takes its expectation value when Q is fixed. Clearly, for the PES at hand this path coincides with MEP, since is a harmonic oscillator. 

The modeling of a groundwater chemical pollution problem may be one-, two-, or tlu-cc-dimcnsional. The proper approach is dependent on the problem context. For c.xamplc, tlie vertical migration of a chemical from a surface source to the water table is generally treated as a one-dimensional problem. Within an aquifer, this type of analysis may be valid if the chemical nipidly penetrates the aquifer so that concentrations are uniform vertically and laterally. This is likely to be the case when the vertical and latcrtil dimensions of the aquifer arc small relative to the longitudinal scale of the problem or when the source fully penetrates the aquifer and forms a strip source. 

Forces are vector quantities and the potential energy t/ is a scalar quantity. For a three-dimensional problem, the link between the force F and the potential U can be found exactly as above. We have... 

These equations assume that the reactor is single phase and that the surroundings have negligible heat capacity. In principle, Equations (14.19) and (14.20) can be solved numerically using the simple methods of Chapters 8 and 9. The two-dimensional problem in r and is solved for a fixed value of t. A step forward in t is taken, the two-dimensional problem is resolved at the new t, and so on. 

For two-dimensional problems, if a bilinear interpolation function is employed, the influence coefficients can be computed likewise in analytical form [31]. 

A direct computation of Eq (27) may reach accuracy up to the level of discrete error, but this needs multiplications plus (N-i) additions. For two-dimensional problem, it needs N XM multiplications and (W-1) X (M-1) additions. The computational work will be enormous for very large grid numbers, so a main concern is how to get the results within a reasonable CPU time. At present, MLMI and discrete convolution and FFT based method (DC-FFT) are two preferential candidates that can meet the demands for accuracy and efficiency. 

A locally one-dlinensional scheme (LOS) for the heat conduction equation in an arbitrary domain. The method of summarized approximation can find a wide range of application in designing economical additive schemes for parabolic equations in the domains of rather complicated configurations and shapes. More a detailed exploration is devoted to a locally one-dimensional problem for the heat conduction equation in a complex domain G = G -f F of the dimension p. Let x — (sj, 2,..., a- p) be a point in the Euclidean space R. ... 

When solving difference boundary-value problems for Poisson s equation in rectangular, angular, cylindrical and spherical systems of coordinates direct economical methods are widely used that are known to us as the decomposition method and the method of separation of variables. The calculations in both methods for two-dimensional problems require Q arithmetic operations, Q — 0 N og. N), where N is the number of the grid nodes along one of the directions. 

---